A nodal collocation approximation for the multidimensional P L equations. 3D applications.

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1 XXI Congreso de Ecuaciones Diferenciaes y Apicaciones XI Congreso de Matemática Apicada Ciudad Rea, 1-5 septiembre 9 (pp. 1 8) A noda coocation approximation for the mutidimensiona P L equations. 3D appications. M. Capia 1, C. F. Taavera 1, D. Ginestar 1, G. Verdú 1 Dpto. de Matemática Apicada, Universidad Poitécnica de Vaencia, C o de Vera 14, E-46 Vaencia. E-mais: tcapia@mat.upv.es, taavera@mat.upv.es, dginesta@mat.upv.es. Dpto. de Ingeniería Química y Nucear, Universidad Poitécnica de Vaencia, C o de Vera 14, E-46 Vaencia. E-mai: gverdu@iqn.upv.es. Paabras cave: Mutidimensiona P L equations, noda coocation method, Lambda Modes transport probem Abstract For the anaysis of reactors with compex fue assembies or fine mesh appications as pin by pin neutron fux reconstruction, the usua approximation of the neutron transport equation by the mutigroup diffusion equation does not provide good resuts. A cassica approach to sove the neutron transport equation is to appy the spherica harmonics method obtaining a finite approximation known as the P L equations. We have deveoped a noda coocation method to discretize the P L equations for mutidimensiona geometries [9, 13, 5, 6] on a rectanguar mesh based on the expansion of the neutronic fuxes in terms of orthonorma Legendre poynomias. In a previous work [6], the performance of the method and the dominant transport Lambda Modes were obtained for some D probems. In this work we compute the dominant transport Lambda modes for 3D neutron transport benchmarks [1]. 1 Introduction For the anaysis of reactors with compex fue assembies or fine mesh appications as pin by pin neutron fux reconstruction, the usua approximation of the neutron transport equation by the mutigroup diffusion equation does not provide good resuts and better approximations are needed. Neutron transport equation is generay discretized using the S L method for the anguar dependency, but this method exhibits ray effects probems [1]. To eiminate these probems, the spherica harmonics method up to a finite order L is appied, eading to the P L approximation to the transport equation. This approximation is we estabished [7] and the exact transport soution is recovered as L goes to infinity. 1

2 M. Capia, C. F. Taavera, D. Ginestar, G. Verdú In three dimensiona geometries (3D), the number of P L equations is L(L+1)/. These equations are compicated and can aso be formuated as second order equations, but the couping invoves not ony the anguar moments but aso mixed spatia derivatives of these moments. This probem ed to propose the simpified P L (SP L ) approximation in the eary 6 s. For the SP L approximation, it is proposed to repace the second derivatives in the one dimensiona panar geometry P L equations by three dimensiona Lapacian operators avoiding then the compexity of the fu spherica harmonics approximation. In the nineties, theoretica basis for the observed accuracy of the SP L equations in the mutidimensiona appications were provided [3], showing that these equations are high order asymptotic soutions of the transport equation when diffusion theory is the eading order approximation. This approximation is cheap from the computationa point of view and it has been successfuy used to improve the diffusion soution in MOX reactors []. Nevertheess, the best soution from the theoretica point of view, is obtained from the P L approximation. Athough its impementation in a computer program is compicated, the arge improvement in the computationa capabiity of the new computers makes it worthwhie to investigate numerica discretizations for the P L equations [8]. In this work, we use a noda coocation method for the discretization of these equations, which is based on the expansion of the fieds in terms of orthonorma Legendre poynomias [5, 6]. In a previous work [6], the performance of the method and the dominant transport Lambda Modes were obtained for some D probems. The principa aim of this work is to test the capabiity of the noda coocation method to treat 3D reaistic reactor probems. We compute the dominant transport Lambda modes for 3D neutron transport benchmarks [1] and we show that the noda coocation method does not require a sma mesh size to obtain accurate resuts, being abe aso to obtain the subcritica modes for the transport probems. The transport equation and the P L equations We wi focus on the eigenvaue probem known as the Lambda Modes transport probem [1], given by Ω Φ( r, Ω,E) + Σ t ( r,e)φ( r, Ω,E) = + 1 λ + χ p (E) 4π de Ω dω Σ s ( r; Ω,E Ω,E)Φ( r, Ω,E ) de νσ f ( r,e ) dω Φ( r, Ω,E ), (1) + where Φ( r, Ω,E) is the anguar neutron fux at ocation r with energy E and direction given by the unit vector Ω; Σ t is the tota cross section; Σ s is the scattering cross section from ( Ω,E ) to ( Ω,E); Σ f is the fission cross section; ν is the average number of neutrons per fission, χ p is the spectrum and λ are the eigenvaues of the probem. In practica appications, to eiminate the dependence on energy in Eq. (1), an energy muti group approximation is used. Ω

3 A noda coocation approximation for P L equations. The spherica harmonics method consists of expanding the anguar neutron fux as Φ g ( r, Ω) + = φ m,g ( r)y m ( Ω), () = m= where g is the energy group index; Y m are the (normaized) spherica harmonics and the neutronic fieds φ m,g are the coefficients of the expansion. It is aso assumed that scattering depends ony on the reative ange, µ = Ω Ω, and the scattering cross section may be expanded as the foowing series of Legendre poynomias + 1 Σ s,gg ( r,µ ) = 4π Σ s,gg ( r)p (µ ). (3) = If x, y and z are the Cartesian coordinates, the foowing compex coordinates are introduced x ±1 = 1 (x ± iy), x = z, (4) Substituting Eqs. () and (3) in Eq. (1) and integrating with respect to Ω dω Y m ( Ω) resuts in the foowing equations for the coefficients or moments φ m : ( (( 1 ) + m + )( + m + 1) 1/ φ m+1 ( ) ) +1 ( m + )( m + 1) 1/ φ m ( + 3)( + 1) x 1 ( + 3)( + 1) x +1 ( (( + 1 ) m)( m 1) 1/ φ m+1 ( ) ) 1 ( + m)( + m 1) 1/ φ m ( + 1)( 1) x 1 ( + 1)( 1) x +1 ( ) ( + m + 1)( m + 1) 1/ φ m ( ) +1 ( + m)( m) 1/ φ m ( + 3)( + 1) x ( + 1)( 1) x + Σ t φ m = Σ s φ m 1 + δ δ m λ νσ f φ, =,1,..., m =,...,+, (5) where δ is the Kronecker deta. In these equations it is understood that terms invoving coefficients φ m with invaid indices and m are zero. To obtain a finite approximation, the series in Eq. () is truncated at some vaue = L and the approximate Eqs. (5) obtained are known as the P L equations. In the foowing we wi ony consider L to be an odd integer. Using the Eqs. (5) with odd index, the foowing reations between odd coefficients and derivatives of even coefficients can be obtained φ m φ m = D C + r= 1 A + φm r +1 r;m ( 1)r x r D C r= 1 A φ m r 1 r;m, x r with = 1,3,...,L, where D = 1/(Σ t Σ s ) and the foowing constants have been introduced A + 1;m = 1 [( + m + )( + m + 1)] 1/, A + ;m = [( + m + 1)( m + 1)]1/, A 1;,m = 1 [( m)( m 1)] 1/, A ;m = [( + m)( m)]1/, A ± +1;m = A± 1;, m, C+ = [( + 3)( + 1)] 1/, C = C

4 M. Capia, C. F. Taavera, D. Ginestar, G. Verdú Eqs. () are compex, but the anguar neutron fux, Φ( r, Ω), is a rea function. Introducing the rea moments ξ m = Re (φ m ) = 1 (φm + ( 1) m φ m ) and η m = Im (φ m ) = 1 i (φm ( 1) m φ m ), the corresponding set of rea reations is obtained from Eqs. (). These reations are generaizations of the Fick s aw. Repacing Eqs. () into Eqs. (5) with even, the foowing diffusive equations for even coefficients, φ m, resut E m =, =,,4,...,L 1, (6) where E m is given by r,r = 1 A + ( 1) r+r r;m A+ r ;+1,m r φ m r r + C + C + D +1 x +1 r x r r,r = 1 ( ( 1) r A+ r;m A r ;+1,m r C + C +1 + ( 1) r A r;m A+ r ; 1,m r C C + 1 A r;m A r ; 1,m r r,r =1 C C 1 φ m r r D +1 x r x r φ m r r ) D 1 x r x r φ m r r D 1 + (Σ t Σ s )φ m δ δ m νσ f φ x r x. r Eqs. (6) are compex, but the diffusive equations for the rea moments ξ m and η m, are given by E m + ( 1) m E m =, and i(e m ( 1) m E m ) =. The discretization of these equations can then be done using a noda coocation method (NCM), based on the expansion of the neutronic fuxes in terms of orthonorma Legendre poynomias in each node, and conditions for the continuity of the soution are required. This method has been previousy used for the neutron diffusion equation in [9] and [13], generaizing the method presented in a previous work [5] for mutidimensiona rectanguar geometries [6]. We can consider zero fux, refective and boundary conditions. These ast conditions are approximated by setting Marshak s conditions [1] dω Y m ( Ω)Φ g ( r, Ω) = for odd, (7) Ω n where n is the outwardy directed unitary norma vector to the externa surface. 3 Numerica Resuts In this Section, we present resuts from a muti group eigenvaue (k eff ) benchmark probem in three dimensiona reactor core configuration. The noda coocation method (NCM) was impemented into a computer code written in FORTRAN 9, which soves the eigenvaue probem for an arbitrary P L (L odd) approximation. This code has been aready appied to severa one and two dimensiona probems in [5, 6] with satisfactory agreements. We consider the 3D benchmark probem proposed by [11] as part of the reactor physics benchmark probems compied and pubished by the Nucear Energy Agency Committe 4

5 A noda coocation approximation for P L equations. on Reactor Physics Benchmarks (NEACRP). The specifications of geometrica structure, dimensions and materias present are briefy described here. We refer to [1] for another input data such as the muti group cross sections and fission spectrum. For this probem, we wi compare the k eff predictions from the noda coocation method, against the Monte Caro code MCNP4C [4] resuts presented in [14]. The benchmark probem considered is a sma core mode of a Fast Breeder Reactor (FBR). The overa dimensions of the system are cm. As it is shown in Fig. 1, the mode is comprised of a fue region, radia and axia bankets and a contro rod region. Vacuum boundary conditions are appied. For this mode, we have considered two cases: Case (1), the contro rods are withdrawn (the contro rod position is fied with Na) and Case (), the contro rods are haf inserted. Four group materia cross sections are utiized [1]. 14 y (cm) 15 z (cm) 14 x (cm) 14 x (cm) Radia banket Axia banket Core Contro rod Contro rod position (Na fied) Figure 1: Configuration of 3D benchmark probem. Dashed ines correspond to the spatia discretization. Tabe 1 presents the noda coocation method (NCM) soutions for k eff eigenvaue using P L (L = 1,3,5) approximations. To perform these cacuations, M = 6 orthonorma Legendre poynomias in the expansion of the fieds are used. Aso, the XYZ cartesian geometry of the reactor was considered, and the spatia discretizacion aong each direction was taken according with the materia zones (see dashed ines in Fig. 1). One advantage of the NCM is that it does not require a sma mesh size to obtain satisfactory resuts. The number of finite eement meshes in X, Y and Z directions is 15, 15 and 4, respectivey and the tota number of nodes in 3D is 9. Notice that in previous cacuations [1, 14] one quarter of the reactor was taken and the reference mesh size adopted was 5 cm, resuting in 588 3D nodes. In Tabe 1, P L resuts are compared against those from MCNP4C resuts reported in [1]. Here, the k eff eigenvaue from P 5 approximation agrees we with the one computed 5

6 M. Capia, C. F. Taavera, D. Ginestar, G. Verdú Method Case 1 (±1σ) Case (±1σ) MCNP4C.9736(±.8).95985(±.8) NCM P NCM P NCM P Tabe 1: k eff predictions for different anguar P L approximations compared to MCNP4C estimates for the 3D benchmark probem. with MCNP4C, being the difference 15.9 pcm for Case 1, and pcm for Case. We observe that P 5 anguar approximations are sufficient to obtain accurate transport soutions for this k eff eigenvaue probem and aso the spatia mesh used provides satisfactory resuts. In Fig. we pot the four group scaar fuxes obtained from the noda coocation method for Case (contro rod haf inserted), from core center of coordenate (7,7,74.5) to (14,7,74.5). Resuts are normaized such that the tota power is equa to the core voume P 1 P 3 P P 1 P 3 P 5 Scaar Fux Scaar Fux (a) x(cm) (b) x(cm) 3 5 P 1 P 3 P P 1 P 3 P 5 Scaar Fux 15 1 Scaar Fux (c) x(cm) (d) x(cm) Figure : Scaar fux for P L (L = 1,3,5) approximations from core center of coordenate (7,7,74.5) to (14,7,74.5) for Case of benchmark probem. (a) corresponds to group 1, (b) group ; (c) group 3; (d) group 4. 6

7 A noda coocation approximation for P L equations. In order to study the convergence of the k eff resuts obtained using the noda coocation method, we have soved the eigenvaue probem for Cases 1 and by varying the number of Legendre poynomias (M) in the expansion of the scaar fux. The resuts are shown in Tabes and 3. M P 1 P 3 P Tabe : k eff eigenvaue soutions for different vaues of M and different P L approximations (Case 1). M P 1 P 3 P Tabe 3: k eff eigenvaue soutions for different vaues of M and different P L approximations (Case ). Another advantage of the noda coocation method is that we can aso obtain the subcritica modes, as shown in Tabe 4 for the P 5 approximation for this particuar probem. Eigenvaues P 5 (Case 1) P 5 (Case ) k eff nd rd th Tabe 4: Four dominant eigenvaues for the P 5 approximation for the benchmark probem (Case 1 and ). Acknowedgement Partia support to perform this work has been received from the Spanish Ministerio de Educación y Ciencia under project ENE8-669/CON. Bibiography [1] R. E. Acouffe, E. W. Larsen, W. F. Mier Jr., B. R. Wienke, 1979, Computationa efficiency of the numerica methods for the mutigroup discrete ordinates neutron transport equations: The sab geometry 7

8 M. Capia, C. F. Taavera, D. Ginestar, G. Verdú case, Nuc. Sci. Eng. 71 (1979), 111. [] C. Beckert, U. Grundmann, U., Deveopment and verification of a noda approach for soving the mutigroup SP 3 equations, Ann. Nuc. En. 35 (8), [3] P. S. Brantey, E. W. Larsen, The simpified P 3 approximation, Nuc. Sci. Eng. 134 (), 1 1. [4] J. F. Briesmeister, MCNP A genera Monte Caro N partice transport code, Version 4C, Technica Report LA 1379 M, Los Aamos Nationa Laboratory. [5] M. Capia, C. F. Taavera, D. Ginestar, D., G. Verdú. A noda coocation method for the cacuation of the ambda modes of the P L equations. Ann. Nuc. En. 3 (5), [6] M. Capia, C. F. Taavera, D. Ginestar, G. Verdú, A noda coocation approximation for the muti dimensiona P L equations D appications, Ann. Nuc. En. 35 (8), [7] M. Cark Jr., K. F. Hansen, Numerica Methods of Reactor Anaysis, New York Academic, New York, [8] J. K. Fetcher, A soution of the neutron transport equation using spherica harmonics, J. Phys. A: Math. Gen. 16 (1983), [9] A. Hébert. Deveopment of the noda coocation method for soving the neutron diffusion equation. Ann. Nuc. En., 14(1) (1987), [1] W. M. Stacey, W, Nucear Reactor Physics, Wiey, New York,1 [11] T. Takeda, M. Tamitani, H. Unesaki, Proposa of 3-D neutron transport benchmark probems, Technica Report A 953 (1988), Rev. 1. [1] T. Takeda, H. Ikeda, 3-D neutron transport benchmarks, Technica Report OECD/NEA (1991). Committee on Reactor Physics (NEACRP). [13] G. Verdú, D. Ginestar, V. Vida, V., J. L. Muñoz Cobo. λ modes of the neutron diffusion equation. Ann. Nuc. En. 1(7) (1994), [14] A. K. Ziver, M. S. Shahdatuah, M. D. Eaton, C. R. E. Oiveira, C. C. Umpeby, A. J. H. Goddard, Finite eement spherica harmonics (P N) soutions of the three dimensiona Takeda benchmark probems, Ann. Nuc. En. 3 (5),